Time-dependent metric for the two dimensional, non-Hermitian coupled oscillator

Abstract

We provide a time-dependent Dyson map and metric for the two dimensional harmonic oscillator with a non-Hermitian $ixy$ coupling term. This particular time-independent model exhibits spontaneously broken $\mathcal{P}\mathcal{T}$-symmetry and becomes unphysical in the broken regime, with the spectrum becoming partially complex. By introducing an explicit time-dependence into the Dyson map, we provide a time-dependent metric that renders the model consistent across the unbroken and broken regimes.

7 pages